Sprint Corp. board members Masayoshi Son and Dan Hesse met recently with Justice Department officials who said they would view a Sprint acquisition of wireless rival T-Mobile US Inc. with skepticism, people briefed on the conversation said.

The conversation, which occurred in January, shows the seriousness of Mr. Son's interest in a deal, and his highest hurdle. U.S. antitrust authorities regard the current lineup of four national mobile-phone carriers as important to maintaining a competitive market, and department...